In this work we introduce and analyse a new adaptive Petrov-Galerkin heterogeneous multiscale finite
element method (HMM) for monotone elliptic operators with rapid oscillations.
In a general heterogeneous setting we prove convergence of the
HMM approximations to the solution of a macroscopic limit equation.
The major new contribution of this work is an a-posteriori error estimate
for the $L^2$-error between the HMM approximation and the solution of the
macroscopic limit equation.
The a posteriori error estimate is obtained in a general heterogeneous setting
with scale separation without assuming periodicity or stochastic ergodicity.
The applicability of the method and the usage of the a posteriori error estimate
for adaptive local mesh refinement is demonstrated in numerical experiments.
The experimental results underline the applicability of the a posteriori error
estimate in non-periodic homogenization settings.